Corners and Stars

In the former image, we have an octagon at the 3x3 square grid. In the latter image, we have a regular octagon. They are not the same.
The octagon at the 3x3 grid has four symmetry axes. It has a 90° rotational symmetry. There are two different side lengths, whose ratio is $\sqrt{\text{2}}$. The height is three times the shorter side lengths. The area is 7/9 of the area of the square frame.

The two pentagons are irregular. They are convex and symmetric. They both have three 90° angles and two 135° angles.
The pentagon at the 2x2 square grid has three side lengths that are proportional to $\text{1}$, $\text{2}$, $\sqrt{\text{2}}$.
The pentagon from the Vichten mosaic has three side lengths that are proportional to $\text{1}$, $\text{2}$, $\text{1 +}\sqrt{\text{2}}$ (it may help to see the pentagon as a quarter of a regular octagon).

The two stars consist of selected diagonals of the regular octagon. The 8/2 star is given by the eight smallest diagonals, among vertices with 2 sides in between. The 8/3 star is given by the eight intermediate diagonals, among vertices with 3 sides in between.

Starting from an 8/2 star, prolonge the eight lines. You get eight additional intersection points which are the vertices of an 8/3 star.
Starting from an 8/3 star, the segments in the interior of the star form an 8/2 star.